#### Vol. 15, No. 3, 2021

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The Hodge ring of varieties in positive characteristic

### Remy van Dobben de Bruyn

Vol. 15 (2021), No. 3, 729–745
##### Abstract

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers ${h}^{i,j}\left(X\right)=dim\phantom{\rule{0.3em}{0ex}}{H}^{j}\left(X,{\Omega }_{X}^{i}\right)$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of $X$ are given by the span of the ${h}^{i,0}\left(X\right)$ and the ${h}^{0,j}\left(X\right)$ (and their duals ${h}^{i,n}\left(X\right)$ and ${h}^{n,j}\left(X\right)$). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.

##### Keywords
algebraic geometry, positive characteristic, Hodge cohomology, de Rham cohomology, Grothendieck ring of varieties, birational invariants
##### Mathematical Subject Classification 2010
Primary: 14G17
Secondary: 14A10, 14E99, 14F40, 14F99